Title: The Holographic Origin of Active Geometry: Endogenous Metric Deformation

This work presents a unified field equation on the principal bundle of oriented frames P = SO (M), resolving the dualism between macroscopic kinematics and microscopic structural reconfiguration. We treat the metric structure not as a fixed background, but as an endogenous degree of freedom capable of active reconfiguration. Key Innovations and Results: Holographic Metric Determination: Macroscopic geometry emerges as a holographic imprint of the microscopic algebra of constraints. Active degrees of freedom are dynamically filtered through an intersection of operator cones, generating the metric tensor G (X) from the local structural potential JIR. Active Geometrodynamics: Metric curvature arises as an endogenous nonlinear deformation induced by the Hessian of the potential. This process transforms topological barriers into permeable corridors through a spectral reconfiguration of mobility. Metric Short-Circuiting (Theorem 2): Utilizing Yosida regularization, we demonstrate how frame orthogonality initiates the annihilation of the effective interval (ds²ₑff → 0), topologically regularizing singularities. Phase Accumulation Law (Theorem 1): We derive the dynamics of reconfiguration from a variational balance principle of least geometric resistance. Empirical Validation: The model confirms the regime of Harmonic Resilience (γ ≈ 2), demonstrating that active systems can undergo reversible metric deformations where passive media would experience brittle failure. Keywords: Unified Field Equation, Active Geometrodynamics, Endogenous Metric Deformation, Holographic Metric Determination, Information–Metric Coupling, Micro–Macro Dynamics, Yosida Regularization, Convex Cone Intersection, Hessian Geometry, Critical Metric Corridor (CMC), Metric Short-Circuiting, Orthogonal Resonance, Topological Phase Transition, Harmonic Resilience.